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Daniell Integral University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1965 Daniell integral Eddward Melvin Wadsworth The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Wadsworth, Eddward Melvin, "Daniell integral" (1965). Graduate Student Theses, Dissertations, & Professional Papers. 8091. https://scholarworks.umt.edu/etd/8091 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. DMIELL INTEGRAL By EDDWARD MELVIN WADSWORTH Bo Ac College of Great Palls 9 1962 Presented in partial fulfillment of the requirements for the degree of Master of Arts MONTANA STATE UNIVERSITY 1965 Approved by & R C b J i J L — Chairman, Board of Examiners Deang/j^raduate School MAR 22 1965 Date Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38892 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI* O îaM rution Publiahing UMI EP38892 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQ^st* ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 -1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OP CONTENTS PAGE ACKNOWLEDGEMENT S CHAPTER lo THEORY OF THE DANIELL INTEGRAL 1 Basic structures, 1 The class L^. 5 Definition and Properties of I. 9 The completion of L to L^. 12 When is a function in L^? 14 CHAPTER II. MEASURABILITY 20 The cr-algehra of measurable sets. 22 A measure u. 24 Stone * s existence theorem. 27 The class L^. 30 The uniqueness of I and u. 33 CHAPTER III. APPLICATION 35 The Riemann Integral* 36 What is in the cr-algebra of u measurable seta? 38 Lebesgue measure is u measure. 42 The Daniell extension of the Riemann 43 Integral yields the Lebesgue Integral. REFERENCES 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I wish to express my gratitude to Dr. David R. Arterburn for his gracious giving of time and guidance for this project, and also for suggesting this exciting subject. I would also like to thank Dr. William M. Myers who made It possible for me to attend graduate school at Montana State University. Further, I would like to thank Drs, William M. Myers, William R. Ballard, and Randolph H. Jeppesen for reading the manuscript, and the National Science Foundation for support of two summers of study. Eddward Melvin Wadsworth Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I THEORY OP THE DAHIELL INTEGRAL We shall develop the theory of the integral as given originally by Daniell [2] , The structure startswith a positive linear functional defined on a vector lattice of functions, defined subsequently, DEFINITION 1,1, A vector space or linear space, V, over a field P is a set of elements called vectors such that f and g in V implies that f + g is in Vj f in V, a in P implies af in V; and further 1 i) 7 is an Abelian group under addition; ii) for all f and g in V and for all a and b in P, a(f+g)=af+ag, (a+b)f=af+bf, (ab)f=a(bf). If = f , DEFINITION 1,2, Let X be an arbitrary set. Suppose f and g are two real-valued functions on X, Then define f V g = max(f,g) = [(f-g)v^ + g and f A g = min(f,g) = (f + g) - (f v g), where 0 is the function identically zero on all of X, DEFINITION 1,3, Suppose S is a set of real-valued functions on a set X, Then S is called a lattice provided that for all f and g in S, f v g and f A g are in V, 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Further, a vector space V of real-valued functions on a set X over the real field is a vector lattice provided that V is also a lattice, LEMMA 1.4. Suppose L is a vector space of real­ valued functions on X. Then L is a vector lattice provided that for each h in L, h v 0 is in L. PROOF, Suppose f and g are in L. Then -g is in L, so f - g is in L, thus (f - g) v 0 is in L and [(f - g) V q) + g is in L. Hence f v g is in L. Similarly, f + g is in L and f v g is in L, so f /\ g is in L, DEFINITION 1.5* Suppose L is a vector lattice of real-valued functions on X, Then if f is in L define |fI = (f V 0) + (-f V 0), Further, let = f v 0. NOTE 1.6. This implies that If I is in L since by 1.4, (f V 0) is in L and (-f v 0) is in L, Note also that this implies that f"*" is in L for all f in L and that |fj = f*” + ( -f ) ^. DEFINITION 1.7. Suppose V is a vector space over a field F. A linear functional I from V to F is a function defined such that if f and g are in V, and a and b are in F, then I(af + bg) = al(f) + bl(g). If V is a vector lattice of real-valued functions and if f(x) > 0 for all X in X implies 1(f) ^ 0 , then I is said to be positive. Note that this implies that if I is positive and if X in X then 1(f) ^ 1(g). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THEORM 1.8. Suppose L is a vector space of real­ valued functions on X. Then L is a vector lattice iff for every f in 1 , g is in L where g(x) is f(x) if f(x) ^ 0 or -f(x) if f(x) < 0. Further, g = |f| . PROOF. Note that by 1.2, (f v 0)(x) = f(x) if f(x) ^ 0 or 0 if f(x) < 0 . only if: L a vector lattice implies |f| is in where [fj is as given in 1.5. Suppose f is in L and suppose p is in X. If f(p) ^ 0 then -f(p) < 0 and (f V 0)(p) = f(p) and (-f v 0)(p) is 0. Hence k|(P) = f(p) = g(p). If f(p) < 0 then -f(p) > 0 so (f v 0)(p) = 0 and («f V 0)(p) = -f(p). Hence, If I(P) = (p) = S(P)o But by 1.6, IfI is in La and thus g is in L, and g = |f|. if Î g in L implies L a vector lattice. Suppose f is in L„ Then g is in L and f v 0 = &(f + g) is in L and by 1.4, L is a vector lattice. DEFINITION 1.9. Define R*, the extended real number system, to be the real numbers with co and -oo adjoined with the following conventions g if a is in R* but neither oo nor -00 then, (i), a + CD = co + a = 00. a + ( -oo) = « oo + a = -oo O a ( -00) = - 00 if a0 . a (oo ) = oo if a > 0 . (ii). -00 < a <oo. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1 1 1 ). 00 + CO = 00, - œ + («oo) = “00, oo « (-oo) = oo, “00 “ ( 00 ) = =» 00 . (Iv), an infinite sum with one or more terms m and no terms of «oo is equal to oo. We may notice here that in R*, every increasing sequence of real numbers has a limit, where we define lim = oo if the sequence is not bounded, DEFINITION 1,10, Suppose that I is a positive linear functional defined ona vector lattice L of real“valued functions on X, Then I is called a Daniell functional or a Daniell integral if whenever is an increasing sequence of functions in L and f is in L and is such that f(x) ^ lim fj^(x) for all x in X, then 1(f) ^ lim I(f^). LEMMA 1,11, Suppose <ju^is a sequence of non­ negative functions in L, Suppose that I is a Daniell integral and further that g in L is such that g ( x ) < 22 % (^) for all X in X, Then 1(g) <_ ^ I(u^), PROOF, Define f_(x) = ^ u.(x) for all x in X, ^ i=l ^ Then f^ is in L for all n and lim f^(x) = ^ u^(x) for all X in X, Now, g(x) < ^ u (x) which is lim f (x) for n ^ all X in X, Thus 1(g) < lim I(f^), But n + I(u ) = 2 7 .
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